Lemma 1.3 ([1]}) 
If \( B_1 \subset \widetilde{B} \)  are two Banach spaces with compact embedding, and the real numbers \( \alpha \in (0,1) \) , \( p>1 \)  satisfy \( \alpha p > 1 \) ,
then the space \( W^{\alpha ,p}(0,T; B_1) \)  is compactly embedded into \( \mathcal {C}([0,T]; \widetilde{B}) \) .

